Courses.
Dominique Bakry (IMT, LSP, IUF, UPS)
Diffusions, orthogonal polynomials and related topics Lecture Notes |
Boguslaw Zegarlinski (Imperial College)
Coercive Inequalities and Markov Semigroups in Infinite Dimensions |
Talks.
Wednesday
Julián Martínez (University of Buenos Aires, IMAS-CONICET)
Branching Brownian particles with spatial selection and the KPP equation
Abstract: The F-KPP equation was introduced in 1937 as a model for the evolution of a genetic trait. This equation admits an infinite number of travelling wave solutions but only one of them has a physical meaning, the one with minimal velocity. We consider a system of N interacting Branching Brownian particles and show that the empirical cumulative distribution associated to this process converges to the solution of the F-KPP equation. Additionally, for each N, we prove existence of a velocity for the cloud of particles. These velocities turns out to converge to the minimal one for the F-KPP, namely, a "microscopic selection principle" holds.
Jorge Antezana (University of La Plata, IAM-CONICET)
Gaussian analytic functions on Hilbert spaces of entire functions
Abstract: Given an orthonormal basis {e_n} in a certain Hilbert space of entire functions, and i.i.d. random variables {a_n} with real normal distribution N(0,1), we consider the Gaussian Analytic Function (GAF):
F(z)=\sum_n a_n e_n(z).
In this talk, firstly we recall some basic facts about GAFs. Then, I will focus on GAFs defined in the Paley Wiener spaces, as well as, in de Branges spaces. In this setting we will discuss some results about gap probabilities, universality and the relation about the (random) zeros of a GAF and the structure of the underlying Hilbert space. This talk is based on a joint work with Jerry Buckley, Jordi Marzo and Jan-Fredrik Olsen.
Thursday
Daniel Galicer (University of Buenos Aires, IMAS-CONICET)
The minimal volume of simplices containing a convex body
Abstract: An old question in convex geometry is the following:
How small is the the measure of Smin(K), the simplex of minimal volume enclosing a given convex body K ⊂ Rn?
Given a body K in Rn with barycenter at the origin, we show there is a simplex S inside K having also barycenter at the origin with large volume. This is achieved using stochastic geometric techniques. Precisely, if K is in isotropic position, we present a method to find centered simplices with that property that works with extremely high probability. As a consequence, we provide correct asymptotic estimates (when the dimension n goes to in nity) on the aforementioned problem. Up to an absolute constant, the estimate cannot be lessened. Joint work with Damian Pinasco and Mariano Merzbacher.
Pablo Groisman (University of Buenos Aires, IMAS-CONICET) TBA
Spectral properties of submarkovian operators via probabilistic arguments
Abstract: Let L be the generator of a non-lattice one-dimensional Lévy process with finite exponential moments. Given c>0, we look for positive eigenfunctions (and eigenvalues) of L^* + c(d/dx) in the positive semi-line with zero boundary conditions. We will prove that the set of eigenvalues is given by (0, G(c)], where G is the Legendre transform of L. We will interpret this condition and prove the statement by means of Branching Lévy processes. Time permitting we will discuss other ways in which probabilistic arguments can be used to obtain spectral properties. All this stuff is motivated by the study of quasi-stationary distributions.
Juan Giribet (University of Buenos Aires, IAM-CONICET)
Sampling problems and oblique projections
Abstract: Let H be a complex separable Hilbert space and S be a closed subspace of H. Given a positive operator A in L(H), the pair (A,S) is said compatible if there exists a idempotent operator Q such that AQ=Q^*A and R(Q)=S. The notion of compatibility naturally appears in different approximation problems. In this work we present some approximation problems related with signal processing applications, and we show how the notion of compatibility is connected with these problems.
Friday
Inés Armendáriz (University of Buenos Aires, IMAS-CONICET) TBA
Gibbs measures on permutations of point processes
Abstract: Resumen: We consider the space of permutations of point processes in d-dimensional space, and establish sufficient conditions for the existence and uniqueness of Gibbs measures on this space in the high temperature regime, associated to a general potential. We show that these measures are supported on finite cycle permutations. Joint work with P.A. Ferrari and N. Frevenza
Pablo Shmerkin (University Torcuato Di Tella, CONICET)
A relative Szemerédi Theorem for random fractals
Abstract: The famous Szemerédi Theorem asserts that a subset of the integers of positive upper density contains arbitrarily long arithmetic progressions. Many relative versions, in which the integers are replaced by some sparse set, often random, have been obtained in the discrete setting. I will present results in the same spirit in the continuous setting, that is, for random subsets of some Euclidean space. Joint work with Ville Suomala.
Julián Martínez (University of Buenos Aires, IMAS-CONICET)
Branching Brownian particles with spatial selection and the KPP equation
Abstract: The F-KPP equation was introduced in 1937 as a model for the evolution of a genetic trait. This equation admits an infinite number of travelling wave solutions but only one of them has a physical meaning, the one with minimal velocity. We consider a system of N interacting Branching Brownian particles and show that the empirical cumulative distribution associated to this process converges to the solution of the F-KPP equation. Additionally, for each N, we prove existence of a velocity for the cloud of particles. These velocities turns out to converge to the minimal one for the F-KPP, namely, a "microscopic selection principle" holds.
Jorge Antezana (University of La Plata, IAM-CONICET)
Gaussian analytic functions on Hilbert spaces of entire functions
Abstract: Given an orthonormal basis {e_n} in a certain Hilbert space of entire functions, and i.i.d. random variables {a_n} with real normal distribution N(0,1), we consider the Gaussian Analytic Function (GAF):
F(z)=\sum_n a_n e_n(z).
In this talk, firstly we recall some basic facts about GAFs. Then, I will focus on GAFs defined in the Paley Wiener spaces, as well as, in de Branges spaces. In this setting we will discuss some results about gap probabilities, universality and the relation about the (random) zeros of a GAF and the structure of the underlying Hilbert space. This talk is based on a joint work with Jerry Buckley, Jordi Marzo and Jan-Fredrik Olsen.
Thursday
Daniel Galicer (University of Buenos Aires, IMAS-CONICET)
The minimal volume of simplices containing a convex body
Abstract: An old question in convex geometry is the following:
How small is the the measure of Smin(K), the simplex of minimal volume enclosing a given convex body K ⊂ Rn?
Given a body K in Rn with barycenter at the origin, we show there is a simplex S inside K having also barycenter at the origin with large volume. This is achieved using stochastic geometric techniques. Precisely, if K is in isotropic position, we present a method to find centered simplices with that property that works with extremely high probability. As a consequence, we provide correct asymptotic estimates (when the dimension n goes to in nity) on the aforementioned problem. Up to an absolute constant, the estimate cannot be lessened. Joint work with Damian Pinasco and Mariano Merzbacher.
Pablo Groisman (University of Buenos Aires, IMAS-CONICET) TBA
Spectral properties of submarkovian operators via probabilistic arguments
Abstract: Let L be the generator of a non-lattice one-dimensional Lévy process with finite exponential moments. Given c>0, we look for positive eigenfunctions (and eigenvalues) of L^* + c(d/dx) in the positive semi-line with zero boundary conditions. We will prove that the set of eigenvalues is given by (0, G(c)], where G is the Legendre transform of L. We will interpret this condition and prove the statement by means of Branching Lévy processes. Time permitting we will discuss other ways in which probabilistic arguments can be used to obtain spectral properties. All this stuff is motivated by the study of quasi-stationary distributions.
Juan Giribet (University of Buenos Aires, IAM-CONICET)
Sampling problems and oblique projections
Abstract: Let H be a complex separable Hilbert space and S be a closed subspace of H. Given a positive operator A in L(H), the pair (A,S) is said compatible if there exists a idempotent operator Q such that AQ=Q^*A and R(Q)=S. The notion of compatibility naturally appears in different approximation problems. In this work we present some approximation problems related with signal processing applications, and we show how the notion of compatibility is connected with these problems.
Friday
Inés Armendáriz (University of Buenos Aires, IMAS-CONICET) TBA
Gibbs measures on permutations of point processes
Abstract: Resumen: We consider the space of permutations of point processes in d-dimensional space, and establish sufficient conditions for the existence and uniqueness of Gibbs measures on this space in the high temperature regime, associated to a general potential. We show that these measures are supported on finite cycle permutations. Joint work with P.A. Ferrari and N. Frevenza
Pablo Shmerkin (University Torcuato Di Tella, CONICET)
A relative Szemerédi Theorem for random fractals
Abstract: The famous Szemerédi Theorem asserts that a subset of the integers of positive upper density contains arbitrarily long arithmetic progressions. Many relative versions, in which the integers are replaced by some sparse set, often random, have been obtained in the discrete setting. I will present results in the same spirit in the continuous setting, that is, for random subsets of some Euclidean space. Joint work with Ville Suomala.